Particle accelerators can be generally grouped into two basic categories: accelerators which recirculate particles and those which accelerate particles in a line. The latter (linacs) require transverse magnetic fields only to focus or form a beam with finite spatial extent; the net transverse magnetic field at the center of the beam is zero. The first category, however, necessitates the use of a system of guide magnets to confine the beam in a circular or spiral orbit during acceleration. This magnetic field can take many different forms. For example, in the case of a synchrotron, the magnetic field is varied with time to keep charged particles confined to a closed orbit which lies within a very limited radial range as the particles accelerate and gain energy; the beam orbit experiences no or little change in radius with energy. Synchrotrons have small magnetic component apertures which accommodate the transverse size of the circulating beam but the magnetic field must be ramped, or pulsed, in synch with the beam energy. Synchrotrons therefore have limited duty cycle due to the time required to pulse and recycle a magnetic field. A betatron behaves in a similar fashion, requiring the guide field to vary with energy.
In a recirculating accelerator with fixed magnetic fields, no such cycling is necessary, but the position of the particle beam changes as it gains energy. Stable orbits require the integrated magnetic strength to scale with the momentum of the particles, and to accurately track the position of the particle beam as it moves outward across the magnetic aperture during acceleration. In the case of some cyclotrons, the magnet field is very nearly uniform in space and time and particles move into larger orbits as their energies rise, with a correspondingly longer path length in the field (and therefore increased integrated field strength as a function of energy). In the case of a fixed field alternating gradient (FFAG) accelerator, the magnetic field strength at a given point in space does not vary with time, but its spatial variation with radius can be large in order to match the increasing momentum of the particles as they gain energy and their orbits change accordingly; that is, in order to limit the required increase in orbit radius, the magnetic field can increase sharply as a function of radius.
The use of a field profile with strong spatial variation can permit particles of very different energies to coexist in close and pre-determined proximity. For applications requiring acceleration of intense beams within a compact space, this is a desirable property. There are many situations where a small footprint or compact size is desirable or necessary because of limitations in space and/or requirements of portability. A small footprint can also be important to achieve high duty cycle and consequently increased beam intensity with a modest acceleration system. As a result, accelerators using the FFAG principle for the guide magnets have attracted particular attention in commercial and research applications requiring high beam power, high duty cycle, reliability, and precisely controlled beams at reasonable cost. See C. Prior, Editor, ICFA Beam Dynamics Newsletter #43, August 2007, http://www-bd-fnal.gov\icfabd\Newsletter43.pdf; M. K. Craddock, “New Concepts in FFAG Design for Secondary Beam Facilities and Other Applications”, PAC'05, Knoxville, Tenn., USA, 16-20 May 2005, p. 261; Machida, http://hadron.kek.jp/˜machida/mirror/misl/publications20090415. pdf; RACCAM project, http://lpsc.in2p3.fr/service_accelerateurs/raccam.htm; E. Keil, http://keil.home.cern.ch/keil/keil.bib; CONFORM project, U.K. (EMMA and PAMELA), http://www.conform.ac.uk; Proceedings of International Workshops on FFAG Accelerators (FFAG00-FFAG08).
FFAG magnet systems and the resulting accelerators can be divided into scaling and non-scaling types. Scaling FFAGs are characterized by geometrically similar orbits of increasing radius. The magnetic field, both in radial sector designs (Keith R. Symon, A Strong Focussing Accelerator with a DC Ring Magnet, MURA Notes, Aug. 13, 1954; and D. W. Kerst, K. R. Symon, L. J. Laslett, L. W. Jones, and K. M. Terwilliger, “Fixed field alternating particle accelerators”, CERN Symposium Proceedings, v. 1, 1956, p. 366) and in spiral sector designs (D. W. Kerst, “Properties of an Intersecting-Beam Accelerating System”, CERN Symposium Proceedings, v. 1, 1956, pp. 36-39), follows the lawsB∝rkF(θ),B∝rkG(Ψ)where r is the radius and k, the constant field index, is the “scaling” attribute. F(θ) and G(Ψ) are dependent on the chosen design. (In the spiral ridge design, the parameter Ψ is related to the physical angle, θ, in a well understood manner related to design details.) Scaling machines are theoretically designed such that at all energies during the acceleration cycle the particle beam executes a fixed number of betatron oscillations about a reference orbit in the course of a revolution. This condition, referred to as constant betatron tune, helps to insure that the beam can be accelerated without encountering the strong low-order resonances which lead to beam blow-up and eventual loss from particles hitting accelerator walls. Scaling FFAG accelerators directly incorporate high-order multipole fields to achieve this constant tune and, in general, require complex magnet shapes, pole profiles and, in the case of the spiral sector design, elaborate edge shaping.
The non-scaling FFAG was originally conceived as a means for the rapid acceleration of muon beams. This type of design did not attempt to achieve a constant betatron tune, employed only a linear radial dependence for the magnetic field strength (thereby achieving a large dynamic aperture) and aimed for beam stability only during a brief acceleration cycle lasting for tens of orbits and spanning a limited energy range. These muon or rapid acceleration designs utilized rectangular, fixed-field magnets that combined steering and transverse focusing only linear (quadrupole) gradients to monotonically increase the field with radius. However, the non-uniform, energy-dependent betatron tunes resulted in the beam crossing many resonances, thus making this approach untenable for gradual acceleration which requires beam stability over a much larger number of turns.
An innovative non-scaling approach to gradual acceleration was subsequently proposed in which the constant tune feature was successfully combined with the simplicity of linear-field, non-scaling FFAG components. (See US Published Patent Application 2007/0273383, “Tune-stabilized, non-scaling, fixed-field, alternating gradient accelerator”, Johnstone, Carol J.) This non-scaling approach is termed a linear-field, linear-edge FFAG; in it, weak and strong focusing principles (both edge and linear-gradient focusing) are applied to fixed-field combined-function magnets, but now with canted entrance and exit faces, so as to stabilize machine tunes. This stabilization occurs though an appropriate increase in the net or integrated field with radius. However, this linear-field non-scaling FFAG remains limited in terms of machine design and features. Generation of stable tunes requires either the use of large component apertures incompatible with a compact system, or the imposition of a restricted acceleration range. While momentum gains of at approximately 600% have been achieved with this type of design, the approach fails to provide the compactness and expanded tune stability needed for applications requiring larger momentum gains with modest acceleration per turn.